Characterization of the glial scar tissue in a murine model of spinal cord compression, with focus on hyaluronan

Spinal cord injury (SCI) is a devastating neurological disorder that affects thousands of people each year. Although in recent decades significant progress has been made in relation to understanding the molecular and cellular events underlying the nervous damage, spinal cord injury is still a highly disabling condition for which there is no curative therapy. People affected by spinal cord injuries manifested dysfunction or loss, temporary or permanent, of motor, sensory and / or autonomic functions depending on the spinal lesion damaged. Currently, the incidence rate of this type of injury is approximately 15-40 cases per million people worldwide. At the origin of these lesions are: road accidents, falls, interpersonal violence and the practice of sports. In this work we placed the hypothesis that HA is one of the component of the scar tissue formed after a compressive SCI, that it is likely synthetised by the perilesional glial cells and that it might support the permeation of the glial scar during the late phase of SCI. Nowadays, much focus is drawn on the recovery of CNS function, made impossible after SCI due to the high content of sulfated proteoglycans in the extracellular matrix. Counterbalancing the ratio between these proteoglycans and hyaluronic acid could be one of the experimental therapy to re-permeate the glial scar tissue formed after SCI, making possible axonal regrowth and functional recovery. Therefore, we established a model of spinal cord compression in mice and studied the glial scar tissue, particularly through the characterization of the expression of enzymes related to the metabolism of HA and the subsequent concentration thereof at different distances of the lesion epicenter. Our results show that the lesion induced in mice shows results similar to those produced in human lesions, in terms of histologic similarities and behavioral results. but these animals demonstrate an impressive spontaneous reorganization mechanism of the spinal cord tissue that occurs after injury and allows for partial recovery of the functions of the CNS. As regards the study of the glial scar, changes were recorded at the level of mRNA expression of enzymes metabolizing HA i.e., after injury there was a decreased expression of HA synthases 1-2 (HAS 1-2) and an increase of the expression HAS3 synthase mRNA, as well as the enzymes responsible for the HA catabolism, HYAL 1-2. But the amount of HA measured through the ELISA test was found unchanged after injury, it is not possible to explain this fact only with the change of expression of enzymes. At two weeks and in response to SCI, we found synthesized HA by reactive astrocytes and probably by others like microglial cells as it was advanced by the HA/GFAP+ and HA/IBA1+ cells co-location.

Solutions of Tikhonov functional equations and applications to multiplication operators on Szegö spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert-Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.

Complex functions with Geogebra

Complex functions, generally feature some interesting peculiarities, seen as extensions real functions, complementing the study of real analysis. However, the visualization of some complex functions properties requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the study of the functions defined from ℂ to ℂ through traditional techniques and by the use of Domain Colouring. Here, we will show how we can use GeoGebra for the study of complex functions, using several representations and creating tools which complement the tools already provided by the software. Our proposals designed for students of the first year of engineering and science courses can and should be used as an educational tool in collaborative learning environments. The main advantage in its use in individual terms is the promotion of the deductive reasoning (conjecture / proof). In performed the literature review few references were found involving this educational topic and by the use of a single software.

Using GeoGebra to study complex functions

The aim of this workshop to present some of the strategies studied to use GeoGebra in the analysis of complex functions. The proposed tasks focus on complex analysis topics target for students of the 1st year of higher education, which can be easily adapted to pre-university students. In the first part of this workshop we will illustrate how to use the two graphical windows of GeoGebra to represent complex functions of complex variable. The second part will present the use of the dynamic color Geogebra in order to obtain Coloring domains that correspond to the graphic representation of complex functions. Finally, we will use the threedimensional graphics window in GeoGebra to study the component functions of a complex function. During the workshop will be provided scripts orientation of the different tasks proposed to be held on computers with Geogebra version 5.0 or high.